Optimal. Leaf size=44 \[ -\frac{A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{2} B c^2 x^2 \]
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Rubi [A] time = 0.0315554, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ -\frac{A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{2} B c^2 x^2 \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^4} \, dx &=\int \left (c (2 b B+A c)+\frac{A b^2}{x^2}+\frac{b (b B+2 A c)}{x}+B c^2 x\right ) \, dx\\ &=-\frac{A b^2}{x}+c (2 b B+A c) x+\frac{1}{2} B c^2 x^2+b (b B+2 A c) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0272924, size = 43, normalized size = 0.98 \[ A \left (c^2 x-\frac{b^2}{x}\right )+b \log (x) (2 A c+b B)+\frac{1}{2} B c x (4 b+c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 46, normalized size = 1.1 \begin{align*}{\frac{B{c}^{2}{x}^{2}}{2}}+A{c}^{2}x+2\,Bbcx+2\,A\ln \left ( x \right ) bc+{b}^{2}B\ln \left ( x \right ) -{\frac{A{b}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.6958, size = 62, normalized size = 1.41 \begin{align*} \frac{1}{2} \, B c^{2} x^{2} - \frac{A b^{2}}{x} +{\left (2 \, B b c + A c^{2}\right )} x +{\left (B b^{2} + 2 \, A b c\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7248, size = 116, normalized size = 2.64 \begin{align*} \frac{B c^{2} x^{3} - 2 \, A b^{2} + 2 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x \log \left (x\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.354409, size = 42, normalized size = 0.95 \begin{align*} - \frac{A b^{2}}{x} + \frac{B c^{2} x^{2}}{2} + b \left (2 A c + B b\right ) \log{\left (x \right )} + x \left (A c^{2} + 2 B b c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12026, size = 62, normalized size = 1.41 \begin{align*} \frac{1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x - \frac{A b^{2}}{x} +{\left (B b^{2} + 2 \, A b c\right )} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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